Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 5, 1-16;http://www.math.u-szeged.hu/ejqtde/
Multiplicity of solutions for a class of quasilinear elliptic
equations with concave and convex terms in
R
∗Uberlandio B. Severo
†Departamento de Matem´atica
Universidade Federal da Para´ıba 58051-900, Jo˜ao Pessoa - PB - Brazil
Abstract
In this paper the Fountain theorem is employed to establish infinitely many solutions for the class of quasilinear Schr¨odinger equations −Lpu + V(x)|u|p−2u = λ|u|q−2u + µ|u|r−2u in R, where L
pu = (|u′|p−2u′)′ +
(|(u2)′|p−2(u2)′)′u,λ, µare real parameters, 1< p <∞, 1< q < p,r >2p and the potentialV(x) is nonnegative and satisfies a suitable integrability condition.
AMS Subject Classification: 35J20, 35J60, 35Q55.
Key words and phrases: Quasilinear elliptic equation; Schr¨odinger equation; Variational method; Fountain theorem; p-Laplacian.
1
Introduction
In this paper we establish multiple solutions for quasilinear elliptic equations of the form
−Lpu+V(x)|u|p−2u=λ|u|q−2u+µ|u|r−2u, (1.1)
u∈W1,p(R), where
Lpu:= (|u′|p−2u′)′+ (|(u2)′|p−2(u2)′)′u,
λ, µ ∈ R, 1< p < ∞, 1 < q < p and r >2p. We assume that the potential V(x) is nonnegative, locally bounded and satisfies the conditions
∗Research partially supported by Millennium Institute for the Global Advancement of Brazilian
Mathematics-IM-AGIMB.
(V1) For some R0 ∈ (0,21p), there holds V(x) ≥ α > 0 for all x ∈ R such that |x|> R0.
(V2)
Z
|x|≥R0
V(x)−1/(p−1)dx <∞.
We work with the space of functions
X =
u∈W1,p(R) :
Z
R
V(x)|u|p dx <∞
which is a separable and reflexive Banach Space when endowed with the norm
kukp =
Z
R
|u′|p dx+
Z
R
V(x)|u|p dx.
Notice that (1.1) is the Euler-Lagrange equation of the functional
Fλ,µ(u) =
1
pkuk
p +2p−1
p
Z
R
|u′|p|u|p dx− λ
q
Z
R
|u|q dx−µ
r
Z
R
|u|r dx.
The functional Fλ,µ is well defined and of class C1 on the space X (see Lemma 2.2)
and we study the existence of solutions of (1.1) understood as critical points ofFλ,µ.
The next theorem contains our main result:
Theorem 1.1 Under the assumptions (V1) − (V2) and supposing 1 < p < ∞, 1< q < p and r >2p we have
(a) for every µ > 0, λ ∈ R, equation (1.1) has a sequence of solutions (uk) such
that Fλ,µ(uk)→ ∞ as k → ∞.
(b) for every λ > 0, µ ∈ R, equation (1.1) has a sequence of solutions (vk) such
that Fλ,µ(vk)<0 and Fλ,µ(vk)→0 as k → ∞.
After the well-known results of Ambrosetti-Brezis-Cerami [4], problems involving elliptic equations with concave and convex type nonlinearities have been studied by several authors, see for example [2], [6], [9], [11] for semilinear problems and [3], [8], [18] for quasilinear problems.
The study of problem (1.1) was in part motivated by the works of Bartsch-Willem [6], Poppenberg-Schmitt-Wang [16] and Ambrosetti-Wang [5]. In [6], Bartsch and Willem proved similar results to the Theorem 1.1 for the semilinear problem −∆u = λ|u|q−2u+µ|u|r−2u in an open bounded domain Ω ⊂ RN with Dirichlet
constrained minimization argument. Ambrosetti-Wang [5] consider the quasilinear problem −u′′+ [1 +εa(x)]u−k[1 +εb(x)](u2)′′u= [1 +εc(x)]uq whereq >1, k, εare
real numbers and a, b, c are real-valued functions belonging to a certain class S. The authors use a variational method, together with a perturbation technique to prove that there existsk0 >0 such that fork >−k0 anda, b, cbelonging toS, the equation has a positive solution u ∈ H1(R), provided that |ε| is sufficiently small. Equations of type (1.1) were also studied in [1].
The special features of this class of problems in the present paper is that it is defined in R, involves thep-Laplacian operator, the nonlinear term (|(u2)′|p−2(u2)′)′u and concave and convex type nonlinearities. Here, we adapt an argument developed by Poppenberg-Schmitt-Wang [16]. Our main results complement and improve some of their results, in sense that we are considering a more general class of operators and nonlinearities, and we have allowed that the potential V vanishes in a bounded part of the domain, so that our results are new even in the semilinear case because we deal with a more general class of potential. To obtain multiplicity of solutions for (1.1), we will apply the Bartsch-Willem Fountain theorem as well as the Dual Fountain theorem, see [6], [19].
Equations like (1.1) in the case p = 2 model several physical phenomena. More explicitly, solutions of the equation
−∆u+V(x)u−k∆(u2)u=g(u) in RN (1.2)
are related to the existence of standing wave solutions for quasilinear Schr¨odinger equations of the form i∂tz = −∆z +V(x)z − h(|z|2)z −k∆f(|z|2)f′(|z|2)z where
V = V(x), x ∈ RN, is a given potencial, k is a real constant and f, h are real functions. For more details on physical motivations and applications, we refer to [5], [16] and references therein.
There are several recent works on elliptic problems involving versions of (1.2), for instance, see [5], [10], [12], [13], [14], [15]. In [13], by a change of variables the quasilinear problem was transformed to a semilinear one and an Orlicz space framework was used as the working space, and they were able to prove the existence of positive solutions of (1.1) by the mountain-pass theorem. The same method of change of variables was used recently also in [10], but the usual Sobolev space
H1(RN) framework was used as the working space and they studied a different class
of nonlinearities. In [14], the existence of both one-sign and nodal ground states of soliton type solutions were established by the Nehari method. But in these papers, the authors do not deal with the concave and convex case.
Notation. In this paper we make use of the following notation:
• RR denotes the integral on the line real and C, C0, C1, C2, ... denote positive (possibly different) constants.
• For R >0, IR denotes the open interval (−R, R).
• For 1 ≤p≤ ∞, Lp(R) denotes the usual Lebesgue space with norms
kukp =
Z
R
|u|pdx
1/p
, 1≤p < ∞;
kuk∞ = inf{C >0 :|u(x)| ≤C almost everywhere inR}.
• For 1 ≤ p < ∞, W1,p(R) denotes the Sobolev space modeled on Lp(R) with
norm
kuk1,p=
Z
R
(|u′|p+|u|p)dx
1/p
.
• By h·,·i we denote the duality pairing betweenX and its dual X∗;
• We denote weak convergence in X by “⇀” and strong convergence by “→”.
2
Abstract framework
In this section we establish some properties of the space X and functional Fλ,µ. By
the condition (V1), it follows that the embeddingX ֒→W1,p(R) is continuous. Indeed,
kukp1,p ≤
Z
R
|u′|p dx+
Z R0
−R0
|u|p dx+α−1
Z
|x|>R0
V(x)|u|p dx
≤max{1, α−1}kukp+ 2R0kukp∞.
Sincekukp
∞ ≤pkuk
p
1,p , see for instance Brezis [7], we conclude that
kuk1,p≤
max{1, α−1} 1−2pR0
1/p
kuk.
Furthermore, we have the following proposition:
Proposition 2.1 Under the conditions(V1)−(V2), the embedding fromX intoLs(R)
Proof. Using H¨older’s inequality, we get
Z
|x|≥R0
|u| ≤
Z
|x|≥R0
V(x)|u|p dx
1/pZ
|x|≥R0
V(x)−1/(p−1)dx
p/(p−1)
≤Ckuk.
Hence
kuk1 =
Z R0
−R0
|u|dx+
Z
|x|>R0
|u|dx≤2R0kuk∞+Ckuk ≤C1kuk.
Since X is immersed continuously in L∞(R), we can conclude by interpolation that the embedding from X into Ls(R) is continuous for 1 ≤ s ≤ ∞. Now, we prove
the compactness. Let (un) be a sequence in X satisfying kunk ≤ C. Thus, up to a
subsequence,
un⇀ u0 in X.
Without lost of generality, we can assume that u0 = 0. Let us show that un → 0 in
L1(R). Given ǫ >0, for R >0 large enough we obtain
Z
|x|≥R
V(x)−1/(p−1)dx < ǫ
2C
(p−1)/p
.
Hence
Z
|x|≥R
|un|dx≤
Z
|x|≥R
V(x)|un|p dx
1/pZ
|x|≥R
V(x)−1/(p−1)dx
p/(p−1)
< ǫ
2Ckunk ≤ ǫ
2. On other hand, as the embeddingW1,p(I
R)֒→L1(IR) is compact there existsn0 such that for all n≥n0
Z R
−R
|un|dx <
ǫ
2. Thus, for all n≥n0
kunk1 =
Z R
−R
|un|dx+
Z
|x|≥R
|un|dx < ǫ
which implies that un →0 in L1(R). From this convergence, if 1 < s <∞ we obtain
kunkss =
Z
R
|un|s−1|un|dx≤ kunk∞s−1kunk1 ≤Ckunk1 →0 as n→ ∞,
and consequently
un →0 in Ls(R) for 1≤s <∞
Lemma 2.2 The functional Fλ,µ:X →R is of class C1 in X and for u, v ∈X because for the other summands the proof is standard.
Existence of Gateaux derivative: Letu, v ∈X and 06=t∈R. We have follows, from (2.1) and Lebesgue’s dominated convergence theorem, that
hΦ′(u), vi=p
Continuity of the Gateaux derivative: We choose a sequence un →u inX and
According to the Lebesgue’s dominated convergence theorem, the right hand side tends to 0 uniformly for kvk ≤1. Thus Φ′ :X →X∗ is continuous.
Let us consider the condition below:
(A1) the compact group G acts isometrically on the Banach space X = L∞j=0Xj,
the spaces Xj are invariant and there exists a finite dimensional space V such
that, for every j ∈N, Xj ≃V and the action of G on V is admissible, that is, every continuous equivariant map ∂U → Vk−1, where U is an open bounded invariant neighborhood of 0 in Vk, k ≥2, has a zero.
From now on, we follow the notations:
Yk := k
M
j=0
Xj, Zk :=
∞
M
j=k
Xj,
To prove the item (a) of the Theorem 1.1, we shall use the Fountain theorem of T. Bartsch [6] as given in [19, Theorem 3.6]:
Lemma 2.3 (Fountain Theorem) Under the assumption (A1), let I ∈ C1(X,R)
be an invariant functional. If, for every k ∈N, there exist ρk> rk >0 such that
(A2) ak:= max
u∈Yk,kuk=ρkI(u)≤0;
(A3) bk:= min
u∈Zk,kuk=rkI(u)→ ∞, k→ ∞;
(A4) I satisfies the (P S)c condition for every c >0,
then I has an unbounded sequence of critical values.
For the item (b), we shall apply a dual version of the Fountain Theorem, see [6, Theorem 2] or [19, theorem 3.18].
Lemma 2.4 (Dual Fountain Theorem) Under the assumption (A1), let I ∈
C1(X,R) be an invariant functional. Moreover, suppose that I satisfies the following
conditions:
(B1) for every k ≥k0, there exists Rk >0 such that I(u)≥ 0 for every u∈Zk with
kuk=Rk;
(B2) bk:= inf u∈Zk,kuk≤Rk
I(u)→0 as k → ∞;
(B3) for every k ≥ 1 there exists rk ∈ (0, Rk) and dk < 0 such that I(u) ≤ dk for
(B4) every sequence(un)⊂YnwithI(un)<0bounded and(I|Yn)
′(u
n)→0asn → ∞
has a subsequence which converges to a critical point of I.
Then for each k≥k0, I has a critical value ck ∈[bk, dk] and ck →0 as k → ∞.
Observe that (B2) and (B3) imply bk≤dk <0.
3
The Palais-Smale condition
We begin this section by proving that any Palais-Smale sequence of the functional Fλ,µ is bounded.
Lemma 3.1 Any (P S)c sequence in X, that is, satisfying Fλ,µ(un) → c and
F′
λ,µ(un)→0 is bounded.
Proof. We have forn large
Fλ,µ(un)−
1
rhF
′
λ,µ(un), uni ≤ kunk+c+ 1.
On the other hand, by Lemma 2.2,
Fλ,µ(un)−
1
rhF
′
λ,µ(un), uni=
1
p−
1
r
kunkp+ 2p−1
1
p −
2
r
Z
R
|u′n|p|un|p dx
+λ
1
r −
1
q
Z
R
|un|q dx.
Thus,
kunk+c+ 1 ≥
1
p−
1
r
kunkp−Ckunkq,
from which it follows that (un) is bounded since r >2pand p > q.
Lemma 3.2 (The Palais-Smale condition) The functional Fλ,µ satisfies the
(P S)c condition for all c∈R.
Proof. Let (un) be in X satisfying Fλ,µ(un) → c and Fλ,µ′ (un) → 0. We will show
that (un) has a convergent subsequence. By Lemma 3.1, (un) is bounded. Thus, up
to a subsequence, un ⇀ u in X and using Proposition 2.1 un → u in Lq(R) and in
Ls(R). Therefore, the H¨older’s inequality implies that
Z
R
|un|q−2un(un−u) dx→0 and
Z
R
|un|r−2un(un−u) dx→0 as n→ ∞.
We have
n → ∞, the equality above can be rewritten as
where x, y ∈ RN, Cp > 0 andh·,·i is the standard scalar product in RN (see Simon [17]), the second summand inI1 is nonnegative and applying the mean value theorem to the integrand of the first integral, we get
I1 ≥
purpose is to estimate the integralI2. We have that
I2 =
By (3.2), we can conclude that
Writing the integral I4 in the following way
and by the inequality (3.3) we get
2pCpǫ+on(1) ≥
Using the H¨older’s inequality and the fact that (u′
n) is bounded inLp(R) we get
4
Proof of the main result
Now, we are ready to prove the main theorem: Proof of Theorem 1.1.
(a) Let us verify that in Lemma 2.3 the conditions (A1)−(A4) are satisfied. Since
Thus X =L∞j=0Xj where Xj = Rej and on X we consider the antipodal action of
Z2 which verifies the condition (A1). Let us define
βk := sup
because on the finite-dimensional spaceYk all norms are equivalent. Sincer >2pthe
relation (A2) is satisfied for every ρk > rk >0 large enough.
It follows easily from the Proposition 2.1 that αk → 0 as k → ∞. We obtain for observe that on the finite dimensional spaceYk all norms are equivalent. Hence, there
exist C0, C1 >0 such that satisfied. This is precisely the point where λ > 0 enters. Finally, the condition (B4) is showed similarly to Lemma 3.2.
In conclusion, we add some remarks about the behavior of solutions with respect to parameters λ and µ.
Remark 1 (a) For λ ∈ R and µ ≤ 0 there are no solutions with positive energy.
Moreover
inf{kuk:u solves (1.1), Fλ,µ(u)>0} → ∞ as µ→0+.
(b) For µ∈R and λ≤0 there are no solutions with negative energy. Moreover
Proof of (a): Letλ, µ∈R. From F′ and the result follows.
Proof of (b): Fixλ, µ∈R. Similarly, from F′
This implies that forλ ≤0 the only solution with non-positive energy is u = 0. For
µ >0, there are constantsc3, c4 >0 with
c3kukp−λc4kukq≤0,
hence
kukp−q ≤λc4/c3 →0 as λ→0+.
Remark 2 For λ > 0 small, it is easy to see that the functional Fλ,µ possesses the
mountain-pass geometry. Thus, using the mountain-pass theorem (see [19, Theorem 2.10]) equation (1.1) has a mountain-pass type solution. This solution is nonnegative. Indeed, it is enough to work with the functional
Acknowledgements. This paper was written when the author was a PhD candidate at UNICAMP-Universidade Estadual de Campinas-SP-Brazil. The author wishes to express his thanks to professors Djairo G. de Figueiredo and J. M. do ´O.
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